The invention relates, in general, to the field of magnetotelluric (MT) inversion methods, a collection of electromagnetic geophysical prospecting techniques used to inverse image underground conductivity variations. More specifically, the invention provides an apparatus and method that works to minimize resolution loss due to the kind of output stabilization and smoothing commonly used in MT inversion methods. The technique is a particular type of inverse input conditioning that filters out noise effects but, in principle, involves no loss of resolution.
If one considers electromagnetic techniques for imaging underground conductivity variations that employ ideal steady-state, far-field plane wave excitations as input across a range of driving frequencies and restricts attention to inversion methods that are cast in the frequency domain, thenxe2x80x94in the context of practical noise considerationsxe2x80x94one must define all non-steady state and non-plane wave excitations as inadmissible input excitations to the inversion problem. For the purposes of this discussion, the MT inversion problem will be defined by these assumptions.
Every inversion problem involves a model describing all that is assumed about the forward problem of interest. A collection of model pieces in general, this model can be called the prior model because it is based exclusively on what is taken as fact, and to what degree, before any data is observed. No inversion problem is solved or can be solved without a prior model specification, although it is not always set out in an obvious manner. Every inversion problem also involves observed data that may be defined at the output of a forward system, i.e., the input to the inversion process. The prior model includes the specification of such observers. In what follows, an attempt will be made to make the prior specification of the MT problem apparent so as to render clear the contribution of the present invention.
In the ideal noise-free case, the MT forward problem is governed by the steady-state Maxwell""s equations involving plane wave excitation of ground media for a collection of frequencies. The conductivity properties of common ground media require the use of low frequency plane waves to obtain significant depth of penetration. In such media, the required frequencies are typically low enough that, to a good approximation, Maxwell""s equations reduce to diffusion equations and not to wave equations as is more common. This may be emphasized as the reason why MT methods do not enjoy the kind of resolution that is comparable, for example, to radar techniques. Nevertheless, there are existence and uniqueness theorems, e.g., for the one-dimensional inversion problem, that guarantee exact and unique inversion, in principle, for suitably well-behaved conductivity profiles.
It is in the context of this dichotomyxe2x80x94exact, unique inversion is possible in principle, while practical algorithms typically deliver poor resolutionxe2x80x94that the concept of xe2x80x9cill-posednessxe2x80x9d usually arises. Well-posed problems, in particular inversion problems, may be defined as having three properties:
1) A solution exists;
2) The solution is unique; and,
3) The solution depends (Lipschitz) continuously on the data (with a Lipschitz constant that is not too big), i.e., small changes in the input data (small with respect to some input reference) result only in small changes in the solution (small with respect to some output reference).
In the conductivity inversion problem, it is the third condition that presents real difficulty with respect to well-posedness. Indeed, inversion algorithms that do not properly address this third condition often exhibit wild variation in their solution output.
Handling ill-posed problems often involves the use of so-called regularization techniques that essentially xe2x80x9cre-posexe2x80x9d the problem so that all three conditions are satisfied. It is interesting to note that the initial development of regularization theory was motivated by the MT problem itself. Unfortunately, the use of regularization usually costs resolution since dealing with highly variable solutions, i.e., avoiding solutions characterized by high-pass spatial variation, or noise, equivalently amounts to some kind of spatial low-pass filtering. As a result, properly addressing an ill-posed problem, in particular one requiring significant attention to resolution, means that whatever technique is used to render the problem well-posed, it should employ minimum low-pass filtering. Proper address therefore demands a clear definition of an objective component to minimize that can deliver such minimal filtration. In physical problems, such objective functions are ideal when they can be cast directly in terms of the physics of the problem. They are otherwise uncomfortably referred to as ad hoc, though often still necessary for stabilization purposes.
As defined above, the MT inversion problem assumes steady-state plane waves as input. Practically speaking, however, measured electromagnetic fields always have a portion involving time-varying and/or non-planar wave effects. As a result, a central problem is estimating the usable part of the total electromagnetic field on-site, namely, that due to steady-state plane wave excitation and response. Indeed, only this part of the total measured field constitutes physically justifiable input to an MT inversion algorithm proper; the remainder is noise or interference.
Dealing with the steady-state plane wave input requirement involves two basic approaches, one emphasizing source power and the other signal processing. The first concerns the ability of a given source to deliver to the measurement site plane waves of sufficient power, across a broad and dense spectral band, such that any on-site interference is relatively weak in comparison. The second approach emphasizes signal processing methods to derive from the measured signals the maximum content due exclusively to steady-state plane wave input.
Consider the first approach. However powerful the source, wave planarity still depends on justifiable far-field assumptions which in turn depend on the type of source, the source-to-site proximity, and, in the purely spectral approach taken here, the driving frequencies involved. Source types can be divided into natural sources and artificial/man-made sources; the latter can be further broken down according to controlled or uncontrolled sources. Plane waves due to natural sources can be used for MT imaging, but their random nature emphasizes proper signal processing. Some uncontrolled artificial sources offer significant steady-state plane wave power but have a frequency spacing too sparse be used alone. Ground-based controlled sources typically have the problem that either they cannot guarantee the delivery of sufficient power at a measurement site, or, that such a guarantee leads to source-to-site proximities so small as to violate the far-field, plane wave assumption. These difficulties have led to the investigation of controlled source techniques that attempt to include the more complicated near-field model. These methods are therefore not MT techniques and will not be discussed further. More recently, the controlled source problem has been addressed using ionospheric sources that canxe2x80x94by designxe2x80x94reliably deliver steady-state plane waves over global scales. Such sources once again place the emphasis on signal processing techniques to deal with non-plane wave and time-varying noise interference.
Signal processing to address the MT problem relies on the prior model restriction that valid input excitations consist of steady-state plane waves. This means that signal interference for the MT problem as defined consists of:
1) Non-steady state excitations; and,
2) Non-plane wave excitations.
In general, processing field data to filter out steady-state, non-plane wave interference requires the use of both on-site and remote reference sensor measurements at locations far from the primary site. The approach relies on the prior knowledge that such interference cannot be simultaneously far field to well separated locations while steady-state plane waves, by definition, are everywhere far-field. It works as a spatial filter, based on field cross-spectral estimates obtained using measurements at both sites, to deliver impedance ratios that phase compensate for stationary, near-field effects. It should be clear that the strict approach has decreasing performance as the local/remote separation distance gets smaller. The availability of one or more remote references offers improved estimates in the case of natural background noise but, in general, the exploitation of natural plane waves often requires more observation time. It is important to stress that natural background and controlled plane wave sources differ because the latter offer access to excitation priors such as oscillation schedules and direction of arrival information.
At this point it should be stressed that the term xe2x80x9cfar fieldxe2x80x9d is used in two contexts in MT imaging to refer to distances that essentially depend on:
1) Frequency in terms of wavelengths, above ground; and,
2) Both frequency and conductivity in terms of skin depth, below ground.
Above the ground, remote reference distances are assessed in terms of atmospheric wavelengths that depend on the driving frequencies used; the driving frequencies used depend on the both the depth of interest and below the ground conductivity. This chain of dependencies, and the fact that both the depth of interest and the conductivity are usually unknown, makes clear the reason why MT problems in general involve excitation frequencies over very wide bandwidths. As an aside, it can be mentioned that common ground conductivities and depths of interest are the reason why the spectral bands that are used often include low audio frequencies.
A lack of remote reference data requires model priors restricted to time and/or frequency domain characteristics. As an example, consider interference due to large, nearby machinery that cannot be moved. If it is known that the machinery has a regular downtime, then one is clearly motivated to perform measurements during these periods. More sophisticated time domain techniques are of course possible but not discussed here. A common frequency domain technique is based on using coherence statistics, defined in terms of the measured field components point-wise in frequency, to weight, or exclude from further processing, measurements that do not exhibit sufficiently high correlation. This exploits the prior that ideal plane waves have field components that are predictably correlated. For example, the electric and magnetic field components of a plane wave at a given point in free space are precisely 90xc2x0 out-of-phase, in the ideal, and therefore deliver a coherence of unity. It should be obvious that the larger separation scale, the more reliable coherence estimates can be for non-plane wave rejection. On the other hand, electromagnetic fields that do not obey a plane wave relation but which still deliver relatively high coherence, even over significantly long baselines, are clearly possible. Hence, care must be taken since the logic of signal rejection using coherence is not contrapositive: Low coherence certainly implies xe2x80x9cnon-plane wave,xe2x80x9d but high coherence does not certainly imply xe2x80x9cplane wave.xe2x80x9d The use of coherence statistics and remote reference techniques together is clearly motivated.
The prior assumption was that the measured signals are steady state, or, in terms of a statistical model, stationary. Indeed, the discussion above dwelled on statistics defined in the frequency domain, where unless this condition is sufficiently satisfied, the long term averaging that is necessary to obtain impedance estimates, coherence statistics, etc., is not well defined. Regarding such temporal aspects, recall that a steady-state plane wave is a conceptual ideal describing not only a source located xe2x80x9cfar away,xe2x80x9d but also one that was started xe2x80x9clong ago.xe2x80x9d The condition of xe2x80x9cfar awayxe2x80x9d enters so that the arriving waves may be modeled as enjoying the property of spatial planarity, and all the simplifications that follow. The condition of xe2x80x9clong agoxe2x80x9d enters, as usual, so that one may consider arbitrary initial conditions as being so small that they can be ignored. But non-steady state and non-plane wave interference, by definition not xe2x80x9cfar awayxe2x80x9d enough, is also a source of noisy signals that are therefore not xe2x80x9clong agoxe2x80x9d enough with respect to computing reliable frequency domain statistics. In particular, if the nature of the disturbances is such that the total signal is not well modeled as a stationary, ergodic process, xe2x80x9cfor significant periods of time,xe2x80x9d then spectral estimates derived over these periods are not well defined. Practically speaking, estimates based on such non-stationary, non-ergodic processes do not xe2x80x9csettlexe2x80x9d to a mean value in the limit of long term averaging.
Such xe2x80x9ctime-varyingxe2x80x9d interference can be addressed using another physical prior, namely, the fact that the underground conductivity is itself time constant. This time-invariance property of the ground suggests a model for the observed resistivity due to steady-state plane wave excitation as a stationary random process. For example, many kinds of non-far field interference are relatively short-lived in time duration, e.g., that due to relatively nearby lightning strokes. Such non-stationary interferers lead to wide-band changes in resistivity estimates based on short-time spectra. This renders it fairly straightforward to reliably detect such spectra and exclude them from input into final resistivity estimation using a number of signal processing techniques. The successful removal of such outliers yields a signal environment more comfortably modeled as stationary and steady-state.
In the following preferred embodiment section of this application, an additional prior is presented that delivers a new, follow-on approach to preparing the impedance estimates for input into MT inversion. This approach is based on finding an optimal fit, as a function of frequency, through a collection of impedance estimates at each analysis frequency. The objective function defining optimality is derived using physical dispersion priors that are known to hold true in the ideal.
The invention works to ensure that resistivities subject to inversion, namely those derived from electromagnetic measurements taken under steady-state plane wave excitation, satisfy the minimum phase and minimum group phase properties required by the physics of the canonic MT problem. Resistivities without these properties are not valid for MT inversion and will lead to reconstruction artifacts and inversion output instabilities. Such artifacts and instabilities may be handled using regularization in the output domain, but their address in this domain usually leads to a direct loss of resolution. On the other hand, these phase minima requirements lead to no loss of resolution in principle, even though they work to maintain a smooth, stable solution if noise is present. Simply stated, the invention is an efficient technique that helps maximize inversion output resolution that might be otherwise squandered for the sake of stabilizing the final solution. It works by minimizing inversion input phase constraints that are known to be satisfied in the ideal, a priori, for resistivities due to plane wave excitations.